Introduction to group theory

Introduction

Please note: a Statement of Participation is not issued for this course.

This free OpenLearn course, Introduction to group theory, is an extract from the Open University course M208 Pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a second level course that introduces the three main branches of pure mathematics, namely group theory, analysis and linear algebra. Proofs are a vital part of pure mathematics. Students studying M208 are expected to read through a number of proofs to improve understanding of the course material and to develop mathematical skills, including producing convincing arguments and problem solving.

Introduction to group theory consists of material from M208 Unit GTA1, Symmetry, and has five sections in total. You should set aside between three to four hours to study each of the sections; the main content though is in the first three sections, so you may want to spend a bit longer on these three sections. The whole extract should take about 16 hours to study. The extract is a small part (around 5%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

In this extract, the concept of a group is introduced using the idea of symmetry. It is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in the course. However, a few techniques and definitions taught in earlier units in M208 are present in the extract without explanation. 

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M208 video material (although please note that the PDF may contain references to other parts of M208). In this extract, some illustrations have also been removed due to copyright restrictions.

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Section 1 introduces the set of symmetries of a two-dimensional figure using intuitive ideas of symmetry. These symmetries are then viewed as functions that can be combined under composition. The resulting structure has properties known as closure, identity, inverses and associativity.

Section 2 introduces you to an algebraic notation for recording symmetries. Towards the end of the section you will see how to use the notation to calculate composites of symmetries and the inverse of a symmetry.

Section 3 introduces definitions of a group, an Abelian group and order of a group. You will also see how to check the axioms for a group. The examples of groups used are extended to include groups of numbers – the modular arithmetics, the integers and the real numbers.

Section 4 looks at how to prove that some of the properties of groups appearing earlier in the extract are general properties shared by all groups.

Section 5 extends ideas of symmetry to three-dimensions.

This OpenLearn course is an adapted extract from the Open University course M208 Pure mathematics.